Add the following rational expressions. $\dfrac{2z+7}{2z+10}+\dfrac{9}{2z^5}=$
Explanation: We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({2z+10})\cdot({2z^5})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{2z+7}{{2z+10}}+\dfrac{9}{{2z^5}} \\\\ &=\dfrac{(2z+7)\cdot({2z^5})}{({2z+10})\cdot({2z^5})}+\dfrac{9\cdot({2z+10})}{({2z^5})\cdot({2z+10})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{(2z+7)\cdot(2z^5)}{(2z+10)\cdot(2z^5)}+\dfrac{9\cdot(2z+10)}{(2z^5)\cdot(2z+10)} \\\\ &=\dfrac{(2z+7)\cdot(2z^5)+9\cdot(2z+10)}{(2z+10)(2z^5)} \\\\ &=\dfrac{4z^6+14z^5+18z+90}{(2z+10)(2z^5)} \end{aligned}$ In conclusion, $\begin{aligned}\dfrac{2z+7}{2z+10}+\dfrac{9}{2z^5}&=\dfrac{4z^6+14z^5+18z+90}{(2z+10)(2z^5)}\\&=\dfrac{2z^6+7z^5+9z+45}{(2z+10)(z^5)}\end{aligned}$